I would like someone to verify my line of thoughts on this problem. This is based de Montmort's matching problem.
- Recall de Montmort's matching problem from chapter 1: in a deck of $n$ cards labeled $1$ through $n$, a match occurs when the number on the card matches the card's position in the deck. Let $X$ be the number of matching cards. Is $X$ binomial? Is $X$ hypergeometric?
Suppose we are interested in the probability that there is $X=1$ matching card.
The matching card can be $1$ of $n$ cards in the deck. There are $(n-2)!$ ways to assign different non-matching card numbers to the remaining $n-1$ cards.
On the same lines,
Perhaps, the above formulation for the PMF of $X$ is correct. I know, that we are sampling without replacement. However, I am having difficulty expressing this as a hypergeometric story proof.
Example. An urn contains $w$ white balls, and $b$ black balls, and $n$ balls are drawn at random, without replacement. The number of white balls in the sample follows $HGeom(w,b,n)$.